We have continued the kinetic modeling of time-resolved spectroscopic data for hemoglobin measured over a range of fractional ligand dissociation. All of the modeling is based on a kinetic formulation of the two-state allosteric model, extended in various ways to incorporate geminate ligand rebinding and tertiary conformational changes. The first model we have studied requires three spectroscopically distinguishable tertiary states of unliganded hemoglobin subunits, two in the R quaternary structure and one in the T quaternary structure. We have refined the kinetic treatment to allow for a stretched-exponential form for the tertiary conformational relaxation, to which is coupled a non-exponential geminate rebinding process, and have introduced linear free energy relations connecting the quaternary transition rates for tetramers with different numbers of bound ligands and/or different configurations of subunit tertiary states. Fits using this model reproduce the measured ligand rebinding and conformational kinetics very well and predict a difference spectrum between the unliganded T and R quaternary states very similar to that which has been documented in other laboratories. In addition, the resulting kinetic parameters predict equilibrium ligand-binding properties which are consistent with the known properties. We are also studying a "tertiary two-state" model in which individual subunits in both the R and T quaternary structures exist in the same two tertiary conformations, r and t, which are identified with the high and low ligand-affinity states, respectively, of the molecule. The r conformation is stabilized by ligand binding and the R quaternary structure, and the t conformation is stabilized by ligand dissociation and the T quaternary structure. Preliminary results suggest that this model describes the kinetic and equilibrium data quite well using only two distinct species spectra.